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## G = D104⋊7C2order 416 = 25·13

### The semidirect product of D104 and C2 acting through Inn(D104)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D104⋊7C2
 Chief series C1 — C13 — C26 — C52 — D52 — D52⋊5C2 — D104⋊7C2
 Lower central C13 — C26 — C52 — D104⋊7C2
 Upper central C1 — C4 — C2×C4 — C2×C8

Generators and relations for D1047C2
G = < a,b,c | a104=b2=c2=1, bab=a-1, ac=ca, cbc=a52b >

Subgroups: 512 in 62 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C13, C2×C8, D8, SD16, Q16, C4○D4, D13, C26, C26, C4○D8, Dic13, C52, D26, C2×C26, C104, Dic26, C4×D13, D52, C13⋊D4, C2×C52, C104⋊C2, D104, Dic52, C2×C104, D525C2, D1047C2
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C4○D8, D26, D52, C22×D13, C2×D52, D1047C2

Smallest permutation representation of D1047C2
On 208 points
Generators in S208
```(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208)
(1 158)(2 157)(3 156)(4 155)(5 154)(6 153)(7 152)(8 151)(9 150)(10 149)(11 148)(12 147)(13 146)(14 145)(15 144)(16 143)(17 142)(18 141)(19 140)(20 139)(21 138)(22 137)(23 136)(24 135)(25 134)(26 133)(27 132)(28 131)(29 130)(30 129)(31 128)(32 127)(33 126)(34 125)(35 124)(36 123)(37 122)(38 121)(39 120)(40 119)(41 118)(42 117)(43 116)(44 115)(45 114)(46 113)(47 112)(48 111)(49 110)(50 109)(51 108)(52 107)(53 106)(54 105)(55 208)(56 207)(57 206)(58 205)(59 204)(60 203)(61 202)(62 201)(63 200)(64 199)(65 198)(66 197)(67 196)(68 195)(69 194)(70 193)(71 192)(72 191)(73 190)(74 189)(75 188)(76 187)(77 186)(78 185)(79 184)(80 183)(81 182)(82 181)(83 180)(84 179)(85 178)(86 177)(87 176)(88 175)(89 174)(90 173)(91 172)(92 171)(93 170)(94 169)(95 168)(96 167)(97 166)(98 165)(99 164)(100 163)(101 162)(102 161)(103 160)(104 159)
(105 157)(106 158)(107 159)(108 160)(109 161)(110 162)(111 163)(112 164)(113 165)(114 166)(115 167)(116 168)(117 169)(118 170)(119 171)(120 172)(121 173)(122 174)(123 175)(124 176)(125 177)(126 178)(127 179)(128 180)(129 181)(130 182)(131 183)(132 184)(133 185)(134 186)(135 187)(136 188)(137 189)(138 190)(139 191)(140 192)(141 193)(142 194)(143 195)(144 196)(145 197)(146 198)(147 199)(148 200)(149 201)(150 202)(151 203)(152 204)(153 205)(154 206)(155 207)(156 208)```

`G:=sub<Sym(208)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (1,158)(2,157)(3,156)(4,155)(5,154)(6,153)(7,152)(8,151)(9,150)(10,149)(11,148)(12,147)(13,146)(14,145)(15,144)(16,143)(17,142)(18,141)(19,140)(20,139)(21,138)(22,137)(23,136)(24,135)(25,134)(26,133)(27,132)(28,131)(29,130)(30,129)(31,128)(32,127)(33,126)(34,125)(35,124)(36,123)(37,122)(38,121)(39,120)(40,119)(41,118)(42,117)(43,116)(44,115)(45,114)(46,113)(47,112)(48,111)(49,110)(50,109)(51,108)(52,107)(53,106)(54,105)(55,208)(56,207)(57,206)(58,205)(59,204)(60,203)(61,202)(62,201)(63,200)(64,199)(65,198)(66,197)(67,196)(68,195)(69,194)(70,193)(71,192)(72,191)(73,190)(74,189)(75,188)(76,187)(77,186)(78,185)(79,184)(80,183)(81,182)(82,181)(83,180)(84,179)(85,178)(86,177)(87,176)(88,175)(89,174)(90,173)(91,172)(92,171)(93,170)(94,169)(95,168)(96,167)(97,166)(98,165)(99,164)(100,163)(101,162)(102,161)(103,160)(104,159), (105,157)(106,158)(107,159)(108,160)(109,161)(110,162)(111,163)(112,164)(113,165)(114,166)(115,167)(116,168)(117,169)(118,170)(119,171)(120,172)(121,173)(122,174)(123,175)(124,176)(125,177)(126,178)(127,179)(128,180)(129,181)(130,182)(131,183)(132,184)(133,185)(134,186)(135,187)(136,188)(137,189)(138,190)(139,191)(140,192)(141,193)(142,194)(143,195)(144,196)(145,197)(146,198)(147,199)(148,200)(149,201)(150,202)(151,203)(152,204)(153,205)(154,206)(155,207)(156,208)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (1,158)(2,157)(3,156)(4,155)(5,154)(6,153)(7,152)(8,151)(9,150)(10,149)(11,148)(12,147)(13,146)(14,145)(15,144)(16,143)(17,142)(18,141)(19,140)(20,139)(21,138)(22,137)(23,136)(24,135)(25,134)(26,133)(27,132)(28,131)(29,130)(30,129)(31,128)(32,127)(33,126)(34,125)(35,124)(36,123)(37,122)(38,121)(39,120)(40,119)(41,118)(42,117)(43,116)(44,115)(45,114)(46,113)(47,112)(48,111)(49,110)(50,109)(51,108)(52,107)(53,106)(54,105)(55,208)(56,207)(57,206)(58,205)(59,204)(60,203)(61,202)(62,201)(63,200)(64,199)(65,198)(66,197)(67,196)(68,195)(69,194)(70,193)(71,192)(72,191)(73,190)(74,189)(75,188)(76,187)(77,186)(78,185)(79,184)(80,183)(81,182)(82,181)(83,180)(84,179)(85,178)(86,177)(87,176)(88,175)(89,174)(90,173)(91,172)(92,171)(93,170)(94,169)(95,168)(96,167)(97,166)(98,165)(99,164)(100,163)(101,162)(102,161)(103,160)(104,159), (105,157)(106,158)(107,159)(108,160)(109,161)(110,162)(111,163)(112,164)(113,165)(114,166)(115,167)(116,168)(117,169)(118,170)(119,171)(120,172)(121,173)(122,174)(123,175)(124,176)(125,177)(126,178)(127,179)(128,180)(129,181)(130,182)(131,183)(132,184)(133,185)(134,186)(135,187)(136,188)(137,189)(138,190)(139,191)(140,192)(141,193)(142,194)(143,195)(144,196)(145,197)(146,198)(147,199)(148,200)(149,201)(150,202)(151,203)(152,204)(153,205)(154,206)(155,207)(156,208) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208)], [(1,158),(2,157),(3,156),(4,155),(5,154),(6,153),(7,152),(8,151),(9,150),(10,149),(11,148),(12,147),(13,146),(14,145),(15,144),(16,143),(17,142),(18,141),(19,140),(20,139),(21,138),(22,137),(23,136),(24,135),(25,134),(26,133),(27,132),(28,131),(29,130),(30,129),(31,128),(32,127),(33,126),(34,125),(35,124),(36,123),(37,122),(38,121),(39,120),(40,119),(41,118),(42,117),(43,116),(44,115),(45,114),(46,113),(47,112),(48,111),(49,110),(50,109),(51,108),(52,107),(53,106),(54,105),(55,208),(56,207),(57,206),(58,205),(59,204),(60,203),(61,202),(62,201),(63,200),(64,199),(65,198),(66,197),(67,196),(68,195),(69,194),(70,193),(71,192),(72,191),(73,190),(74,189),(75,188),(76,187),(77,186),(78,185),(79,184),(80,183),(81,182),(82,181),(83,180),(84,179),(85,178),(86,177),(87,176),(88,175),(89,174),(90,173),(91,172),(92,171),(93,170),(94,169),(95,168),(96,167),(97,166),(98,165),(99,164),(100,163),(101,162),(102,161),(103,160),(104,159)], [(105,157),(106,158),(107,159),(108,160),(109,161),(110,162),(111,163),(112,164),(113,165),(114,166),(115,167),(116,168),(117,169),(118,170),(119,171),(120,172),(121,173),(122,174),(123,175),(124,176),(125,177),(126,178),(127,179),(128,180),(129,181),(130,182),(131,183),(132,184),(133,185),(134,186),(135,187),(136,188),(137,189),(138,190),(139,191),(140,192),(141,193),(142,194),(143,195),(144,196),(145,197),(146,198),(147,199),(148,200),(149,201),(150,202),(151,203),(152,204),(153,205),(154,206),(155,207),(156,208)]])`

110 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 8A 8B 8C 8D 13A ··· 13F 26A ··· 26R 52A ··· 52X 104A ··· 104AV order 1 2 2 2 2 4 4 4 4 4 8 8 8 8 13 ··· 13 26 ··· 26 52 ··· 52 104 ··· 104 size 1 1 2 52 52 1 1 2 52 52 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

110 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D13 C4○D8 D26 D26 D52 D52 D104⋊7C2 kernel D104⋊7C2 C104⋊C2 D104 Dic52 C2×C104 D52⋊5C2 C52 C2×C26 C2×C8 C13 C8 C2×C4 C4 C22 C1 # reps 1 2 1 1 1 2 1 1 6 4 12 6 12 12 48

Matrix representation of D1047C2 in GL2(𝔽313) generated by

 61 0 0 195
,
 0 195 61 0
,
 1 0 0 312
`G:=sub<GL(2,GF(313))| [61,0,0,195],[0,61,195,0],[1,0,0,312] >;`

D1047C2 in GAP, Magma, Sage, TeX

`D_{104}\rtimes_7C_2`
`% in TeX`

`G:=Group("D104:7C2");`
`// GroupNames label`

`G:=SmallGroup(416,125);`
`// by ID`

`G=gap.SmallGroup(416,125);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-13,103,218,50,579,69,13829]);`
`// Polycyclic`

`G:=Group<a,b,c|a^104=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^52*b>;`
`// generators/relations`

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